ITCS 3153 Artificial Intelligence

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ITCS 3153Artificial Intelligence

• Lecture 9
• Adversarial Search
• Chapter 6

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Games

• Shall we play a game?
• Lets play tic-tac-toe

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Minimax
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What data do we need to play?

• Initial State
• How does the game start?
• Successor Function
• A list of legal (move, state) pairs for each
  state
• Terminal Test
• Determines when game is over
• Utility Function
• Provides numeric value for all terminal states

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Minimax Strategy

• Optimal Stragtegy
• Leads to outcomes at least as good as any other
  strategy when playing an infallible opponent
• Pick the option that most (max) minimizes the
  damage your opponent can do
• maximize the worst-case outcome
• because your skillful opponent will certainly
  find the most damaging move

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Minimax

• Algorithm
• MinimaxValue(n)
• Utility (n) if n is a terminal state
• max MinimaxValue(s) of all successors, s if n
  is a MAX node
• min MinimaxValue(s) of all successors, s if n
  is a MIN node

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Minimax
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Minimax Algorithm

• We wish to identify minimax decision at the root
• Recursive evaluation of all nodes in game tree
• Time complexity O (bm)

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Feasibility of minimax?

• How about a nice game of chess?
• Avg branching 35 and avg moves 50 for each
  player
• O(35100) time complexity 10154 nodes
• 1040 distinct nodes
• Minimax is impractical if directly applied to
  chess

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Pruning minimax tree

• Are there times when you know you need not
  explore a particular move?
• When the move is poor?
• Poor compared to what?
• Poor compared to what you have explored so far

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Alpha-beta pruning
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Alpha-beta pruning

• a
• the value of the best (highest) choice so far in
  search of MAX
• b
• the value of the best (lowest) choice so far in
  search of MIN
• Order of considering successors matters (look at
  step f in previous slide)
• If possible, consider best successors first

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Realtime decisions

• What if you dont have enough time to explore
  entire search tree?
• We cannot search all the way down to terminal
  state for all decision sequences
• Use a heuristic to approximate (guess) eventual
  terminal state

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Evaluation Function

• The heuristic that estimates expected utility
• Cannot take too long (otherwise recurse to get
  answer)
• It should preserve the ordering among terminal
  states
• otherwise it can cause bad decision making
• Define features of game state that assist in
  evaluation
• what are features of chess?

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Truncating minimax search

• When do you recurse or use evaluation function?
• Cutoff-Test (state, depth) returns 1 or 0
• When 1 is returned, use evaluation function
• Cutoff beyond a certain depth
• Cutoff if state is stable (more predictable)
• Cutoff moves you know are bad (forward pruning)

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Benefits of truncation

• Comparing Chess
• Using minimax 5 ply
• Average Human 6-8 ply
• Using alpha-beta 10 ply
• Intelligent pruning 14 ply

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Games with chance

• How to include chance in game tree?
• Add chance nodes

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Expectiminimax

• Expectiminimax (n)
• utility(n) if n is a terminal state
• if n is a MAX node
• if n is a MIN node
• if n is a chance node

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Pruning

• Can we prune search in games of chance?
• Think about alpha-beta pruning
• dont explore nodes that you know are worse than
  what you have
• we dont know what we have
• chance node values are average of successors

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History of Games

• Chess, Deep Blue
• IBM 30 RS/6000 comps with 480 custom VLSI chess
  chips
• Deep Thought design came from Campbell and Hsu at
  CMU
• 126 mil nodes / s
• 30 bil positions per move
• routine reaching depth of 14
• iterative deepening alpha-beta search

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Deep Blue

• evaluation function had 8000 features
• 4000 opening moves in memory
• 700,000 grandmaster games from which
  recommendations extracted
• many endgames solved for all five piece combos

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Checkers

• Arthur Samuel of IBM, 1952
• program learned by playing against itself
• beat a human in 1962 (but human clearly made
  error)
• 19 KB of memory
• 0.000001 Ghz processor

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Chinook, Jonathan Schaeffer, 1990

• Alpha-beta search on regular PCs
• database of all 444 billion endgame positions
  with 8 pieces
• Played against Marion Tinsley
• world champion for over 40 years
• lost only 3 games in 40 years
• Chinook won two games, but lost match
• Rematch with Tinsley was incomplete for health
  reasons
• Chinook became world champion

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Othello

• Smaller search space (5 to 15 legal moves)
• Humans are no match for computers

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Backgammon

• Garry Tesauro, TD-Gammon, 1992
• Reliably ranked in top-three players of world

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Discussion

• How reasonable is minimax?
• perfectly performing opponent
• perfect knowledge of leaf node evaluations
• strong assumptions

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Building alpha-beta tree

• Can we restrict the size of game tree?
• alpha-beta will blindly explore tree in
  depth-first fashion even if only one move is
  possible from root
• even if multiple moves are possible, can we use a
  quick search to eliminate some entirely?
• utility vs. time tradeoff to decide when to
  explore new branches or to stay with what you have

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Metareasoning

• Reasoning about reasoning
• alpha-beta is one example
• think before you think
• think about utility of thinking about something
  before you think about it
• dont think about choices you dont have to think
  about

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Goal-directed reasoning / planning

• Minimax starts from root and moves forward using
  combinatorial search
• What about starting at goal and working backward
• We talked about difficulty of identifying goal
  states in bidirectional search
• We do not know how to combine the two in
  practical way

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